Nuprl Lemma : prime-power-divides-product
∀p:ℕ. (prime(p) ⇒ (∀n:ℕ+. ∀x,y:ℤ. ((¬(p | x)) ⇒ (p^n | (x * y)) ⇒ (p^n | y))))
Proof
Definitions occuring in Statement :
prime: prime(a),
divides: b | a,
exp: i^n,
nat_plus: ℕ+,
nat: ℕ,
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
multiply: n * m,
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat_plus: ℕ+,
not: ¬A,
nat: ℕ,
prop: ℙ,
false: False,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
and: P ∧ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
exp: i^n,
prime: prime(a),
divides: b | a,
subtract: n - m,
subtype_rel: A ⊆r B,
sq_type: SQType(T),
guard: {T},
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
uiff: uiff(P;Q)
Lemmas referenced :
nat_plus_properties,
divides_wf,
istype-void,
exp_wf2,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
istype-le,
primrec-wf-nat-plus,
not_wf,
nat_plus_wf,
prime_wf,
istype-nat,
primrec1_lemma,
mul-one,
itermAdd_wf,
int_term_value_add_lemma,
mul-swap,
mul-commutes,
add-commutes,
exp_step,
decidable__lt,
istype-less_than,
add-associates,
add-swap,
zero-add,
int_subtype_base,
set_subtype_base,
le_wf,
subtype_base_sq,
mul_cancel_in_eq,
exp_wf3,
nequal_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
subtype_rel_self,
iff_weakening_equal,
decidable__equal_int,
multiply-is-int-iff,
intformeq_wf,
itermMultiply_wf,
int_formula_prop_eq_lemma,
int_term_value_mul_lemma,
false_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
thin,
rename,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
sqequalRule,
functionIsType,
inhabitedIsType,
because_Cache,
universeIsType,
dependent_set_memberEquality_alt,
dependent_functionElimination,
natural_numberEquality,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
Error :memTop,
independent_pairFormation,
voidElimination,
multiplyEquality,
functionEquality,
intEquality,
productElimination,
addEquality,
equalityIstype,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
sqequalBase,
equalitySymmetry,
instantiate,
cumulativity,
equalityTransitivity,
hyp_replacement,
imageElimination,
universeEquality,
imageMemberEquality,
pointwiseFunctionality,
promote_hyp
Latex:
\mforall{}p:\mBbbN{}. (prime(p) {}\mRightarrow{} (\mforall{}n:\mBbbN{}\msupplus{}. \mforall{}x,y:\mBbbZ{}. ((\mneg{}(p | x)) {}\mRightarrow{} (p\^{}n | (x * y)) {}\mRightarrow{} (p\^{}n | y))))
Date html generated:
2020_05_19-PM-10_02_05
Last ObjectModification:
2020_01_04-PM-08_10_12
Theory : num_thy_1
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